Math's Fundamental Flaw

I'm not sure if it is what you want, but Intuitionism [1] is one area that challenges modern fashions in Mathematical thinking. It suffered greatly under the formalist approach lead by David Hilbert and still has little main-stream support despite Gödel and his incompleteness proofs. Veritasium's latest video on Gödel's incompleteness [2] gives a pretty fair account of how we settled on the current fashionable foundations of Math (including nods to Cantor and Hilbert). For a more formal history there is a book "The Philosophy of Set Theory" [3] that sketches out how we got to where we are. I have always been unsatisfied with the current foundations of math and their obtuse basis in Set Theory. Although I must admit, Category Theory has alleviated that quite a lot, especially with the relaxing of equivalence compared to equality. 1. [link] 2. ▶ this video 3. [link]

in “How inevitable is the concept of numbers?” · View on HN

I think most here would know that math is not complete, consistent or decidable. ( ▶ this video ) But I'm going to leave that aside as it's pretty high level math for me and I never run into those problems in my life. My personal problem with math that prevents me from seeing it as "discovery of fundamental principles claimed to exist in an objective reality" is natural numbers. It's impossible for me to clearly find the number 1 (for example) anywhere. The boundaries between one unit and 0.X or 1.Y seem arbitrary and chosen to help us create models. Religion and philosophy deal with this, but it's also apparent in digital signal processing. Or medicine: are the numerous bacteria in our bodies us, or not? Is the heat radiation from my body me? What about the fact that 1 + 1 rarely if ever equals 2? Meaning that two things together physically interact to create more than the sum of both parts. From celestial bodies to a replicated database the complexity goes through the roof when you have more than one thing.

in “Intuitionism” · View on HN

I know nothing of OP, or the professor he talked to, but I think they might be talking about the higher level concepts of math. For example, set theory. If set theory allowed self reference you could have set R, a set of sets that do not contain themselves. Would R contain R? It can’t, but it can’t not either. It’s self contradictory. The solution, reached through consensus, was to restrict the definition of the set to not allow sets to reference themselves [1]. [1] ▶ this video

in “Computer proof ‘blows up’ centuries-old fluid equations” · View on HN

Going through a shitty phase (which happens often) I've been trying to cope in various ways, reaching high and low, inside and outside, for meaning and purpose and guidance. I don't subscribe to most existing models, not out of a rebellious choice but because I cannot find or follow their logic to a satisfactory conclusion. So far this is what I have, make of it what you will: • There must be a default state, and a part of it must reside in everything built upon it. It doesn't make sense that something began from Nothing, but I like to think that something began from Everything. • But it doesn't mean that a superset system can control a subset system 100% – Think of our relationship with computers and the programs running on/in them, along with all the physics involved and the virtual results. • Veritasium's video on "Math's Fundamental Flaw" [0] was a good source of inspiration: Shit may be coded with hard immutable logic but still have virtually infinite possibilities. • If you think your mind, or your "will" or "heart" or "soul", matters in any way and can make any difference, try to find the cutoff point between -where- that metaphysical quality starts , what sets "you" apart from everything else: What's the difference between you and every other person's mind/soul? Or a person's mind and a computer program? The difference between a world with no humans in it, and a world with 1 human? A barren asteroid and a planet with life? Can a rock access the mysteries of the Univer

in “Living Like Nothing Matters” · View on HN

I found this nice presentation about Godel's theorems. ▶ this video "There is no proof for the statement with Godel-number g". What I'm struggling with is trying to understand if self-reference can be formally represented by some symbol. Can it? Godel's proof seems to rely on a trick that allows a formal system to talk about itself but is that possible? There would need to be a symbol for that and thus a symbol that refers to itself? A picture of a pipe is not a pipe. Similarly a representation of a proof is not a proof, is it?. Not saying I believe I'm right and Godel wrong, but it just feel strange to me that a formal system could ever "prove" anything about itself. What does it mean that something is "true"? That it can be proven?

in “New math book rescues landmark topology proof” · View on HN

This was a great video. But the one titled "Math Has a Fatal Flaw"[1] was even more interesting: the meta-mathematics, thus mathematics of entire mathematics (Gödel). [1] ▶ this video

in “The 3x+1 Problem [video]” · View on HN

Hi YeGoblynQueenne, I always read your logicy comments and posts--thank you! > To clarify, that's incompleteness. Unsoundness is when an inference rule returns wrong results. Incompleteness is where it doesn't return all correct results. As you say, unification without an occurs check is unsound and Prolog's depth-first search is incomplete. Thank you. Yes, I misspoke. For anyone interested, this video gives a nice overview of soundness and completeness, in the context of logic: ▶ this video > I wanted to ask, because you brought it up in the interview, how does minikanren (or minikanrens?) handle the occurs check? You said that you are interested in trying to push the relational aspect as far as possible, recognising that this gives up efficiency - can you quantify to what extent minikanren gives up efficiency and generally say a bit more about the trade-off between (if I understand it correctly) efficiency and relational (declarative?) purity? miniKanren performs unification with the occurs check. This is one of the tradeoffs miniKanren makes: sound unification, but it can be expensive. The interleaving complete search is another tradeoff: the search uses more time and memory than does depth-first-search, in those cases in which DFS would be sufficient. Another tradeoff is that miniKanren avoids the use of non-logical / extra-logical operators. There are no cuts, projected variables, var tests, etc., in the pure parts of the language (which is the version I always use). If you w

in “William Byrd on logic and relational programming, miniKanren (2014)” · View on HN

This was a pair of videos in Veritasium recently - How An Infinite Hotel Ran Out Of Room - [link] and Math Has a Fatal Flaw - ▶ this video .

in “Hilbert's Paradox of the Grand Hotel” · View on HN

If you are mathematically-inclined and are looking forward to understanding the proof itself, the book won't do much better than the following video [1] taken from a comment in this thread [2]. Disappointed, I attempted to read Godel's original papers. That has it's own challenge as it refers to Principia Mathematica. I started reading that. Only to fimd that it uses older mathematical notation not in use today. In short, I haven't found any source that explains the proof in reasonable detail. If you want to understand the thought process of axiomatic reasoning (and with less extrapolations to consciousness et al) it is a good book. Note: I had read a long time back, so do not remember the details. [1] ▶ this video [2] [link]

in “1931: Kurt Gödel shows limits of math, logic, computing, AI” · View on HN